Emerging fossil energy power generation systems must operate with unprecedented efficiency and near-zero emissions, while optimizing profitably amid cost fluctuations for raw materials, finished products, and energy. To help address these challenges, the fossil energy industry will have to rely increasingly on the use advanced computational tools for modeling and simulating complex process systems. In this paper, we present the computational research challenges and opportunities for the optimization of fossil energy power generation systems across the plant lifecycle from process synthesis and design to plant operations. We also look beyond the plant gates to discuss research challenges and opportunities for enterprise-wide optimization, including planning, scheduling, and supply chain technologies.